Optimal. Leaf size=38 \[ \frac {2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {2} x\right ),-1\right )}{\sqrt {3}}+\sqrt {\frac {2}{3}} \sqrt {1-4 x^4} x \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {248, 195, 221} \[ \sqrt {\frac {2}{3}} \sqrt {1-4 x^4} x+\frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {2} x\right )\right |-1\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 221
Rule 248
Rubi steps
\begin {align*} \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx &=\int \sqrt {6-24 x^4} \, dx\\ &=\sqrt {\frac {2}{3}} x \sqrt {1-4 x^4}+4 \int \frac {1}{\sqrt {6-24 x^4}} \, dx\\ &=\sqrt {\frac {2}{3}} x \sqrt {1-4 x^4}+\frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {2} x\right )\right |-1\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 22, normalized size = 0.58 \[ \sqrt {6} x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};4 x^4\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {4 \, x^{2} + 2} \sqrt {-6 \, x^{2} + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {4 \, x^{2} + 2} \sqrt {-6 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 75, normalized size = 1.97 \[ -\frac {\sqrt {-6 x^{2}+3}\, \sqrt {2}\, \sqrt {2 x^{2}+1}\, \left (-12 x^{5}+3 x +\sqrt {2}\, \sqrt {3}\, \sqrt {-6 x^{2}+3}\, \sqrt {2 x^{2}+1}\, \EllipticF \left (\sqrt {2}\, x , i\right )\right )}{9 \left (4 x^{4}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {4 \, x^{2} + 2} \sqrt {-6 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {4\,x^2+2}\,\sqrt {3-6\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \sqrt {6} \int \sqrt {1 - 2 x^{2}} \sqrt {2 x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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